Integration of $e^{ax}cos^n bx$ and $e^{ax}sin^n bx$
Q: Integration of $e^{ax}cos^n bx$ and $e^{ax}sin^n bx$
I know how to integration $e^{ax}cos bx$ using $cos bx=frac{e^{ibx}+e^{-ibx}}{2}$.Using the same trick here I got $$int e^{ax}cos^n bx ~dx=int e^{ax}left(frac{e^{ibx}+e^{-ibx}}{2}right)^n~ dx$$But it doesn't look easy to solve.I google it and find the result only which is:
$$int e^{ax}cos^n bx ~dx=frac{bnsin (bx)+acos (bx)}{a^2+b^2n^2}e^{ax}+frac{b^2(n-1)nint e^{ax}cos^{n-2}bxdx}{a^2+b^2n^2}.$$
Can Anyone help me to figure out this. Any hints or solution will be appreciated.
Thanks in advance.
complex-integration reduction-formula
|
show 2 more comments
Q: Integration of $e^{ax}cos^n bx$ and $e^{ax}sin^n bx$
I know how to integration $e^{ax}cos bx$ using $cos bx=frac{e^{ibx}+e^{-ibx}}{2}$.Using the same trick here I got $$int e^{ax}cos^n bx ~dx=int e^{ax}left(frac{e^{ibx}+e^{-ibx}}{2}right)^n~ dx$$But it doesn't look easy to solve.I google it and find the result only which is:
$$int e^{ax}cos^n bx ~dx=frac{bnsin (bx)+acos (bx)}{a^2+b^2n^2}e^{ax}+frac{b^2(n-1)nint e^{ax}cos^{n-2}bxdx}{a^2+b^2n^2}.$$
Can Anyone help me to figure out this. Any hints or solution will be appreciated.
Thanks in advance.
complex-integration reduction-formula
That "result" is a reduction formula, proved by integrating by parts.
– Lord Shark the Unknown
Nov 24 at 6:16
You can expand your formula by the binomial theorem and integrate termwise.
– Lord Shark the Unknown
Nov 24 at 6:17
If i use binomial theorem then it will consume a lot of time.Is there any other way to figure out this @Lord Shark the Unknown Sir
– emonhossain
Nov 24 at 6:18
As alreasy said, integrate by parts to get the reduction formula.
– Claude Leibovici
Nov 24 at 7:00
You may look at the method I used here.
– Tianlalu
Nov 24 at 7:09
|
show 2 more comments
Q: Integration of $e^{ax}cos^n bx$ and $e^{ax}sin^n bx$
I know how to integration $e^{ax}cos bx$ using $cos bx=frac{e^{ibx}+e^{-ibx}}{2}$.Using the same trick here I got $$int e^{ax}cos^n bx ~dx=int e^{ax}left(frac{e^{ibx}+e^{-ibx}}{2}right)^n~ dx$$But it doesn't look easy to solve.I google it and find the result only which is:
$$int e^{ax}cos^n bx ~dx=frac{bnsin (bx)+acos (bx)}{a^2+b^2n^2}e^{ax}+frac{b^2(n-1)nint e^{ax}cos^{n-2}bxdx}{a^2+b^2n^2}.$$
Can Anyone help me to figure out this. Any hints or solution will be appreciated.
Thanks in advance.
complex-integration reduction-formula
Q: Integration of $e^{ax}cos^n bx$ and $e^{ax}sin^n bx$
I know how to integration $e^{ax}cos bx$ using $cos bx=frac{e^{ibx}+e^{-ibx}}{2}$.Using the same trick here I got $$int e^{ax}cos^n bx ~dx=int e^{ax}left(frac{e^{ibx}+e^{-ibx}}{2}right)^n~ dx$$But it doesn't look easy to solve.I google it and find the result only which is:
$$int e^{ax}cos^n bx ~dx=frac{bnsin (bx)+acos (bx)}{a^2+b^2n^2}e^{ax}+frac{b^2(n-1)nint e^{ax}cos^{n-2}bxdx}{a^2+b^2n^2}.$$
Can Anyone help me to figure out this. Any hints or solution will be appreciated.
Thanks in advance.
complex-integration reduction-formula
complex-integration reduction-formula
edited Dec 4 at 4:19
clathratus
3,017330
3,017330
asked Nov 24 at 6:14
emonhossain
33611
33611
That "result" is a reduction formula, proved by integrating by parts.
– Lord Shark the Unknown
Nov 24 at 6:16
You can expand your formula by the binomial theorem and integrate termwise.
– Lord Shark the Unknown
Nov 24 at 6:17
If i use binomial theorem then it will consume a lot of time.Is there any other way to figure out this @Lord Shark the Unknown Sir
– emonhossain
Nov 24 at 6:18
As alreasy said, integrate by parts to get the reduction formula.
– Claude Leibovici
Nov 24 at 7:00
You may look at the method I used here.
– Tianlalu
Nov 24 at 7:09
|
show 2 more comments
That "result" is a reduction formula, proved by integrating by parts.
– Lord Shark the Unknown
Nov 24 at 6:16
You can expand your formula by the binomial theorem and integrate termwise.
– Lord Shark the Unknown
Nov 24 at 6:17
If i use binomial theorem then it will consume a lot of time.Is there any other way to figure out this @Lord Shark the Unknown Sir
– emonhossain
Nov 24 at 6:18
As alreasy said, integrate by parts to get the reduction formula.
– Claude Leibovici
Nov 24 at 7:00
You may look at the method I used here.
– Tianlalu
Nov 24 at 7:09
That "result" is a reduction formula, proved by integrating by parts.
– Lord Shark the Unknown
Nov 24 at 6:16
That "result" is a reduction formula, proved by integrating by parts.
– Lord Shark the Unknown
Nov 24 at 6:16
You can expand your formula by the binomial theorem and integrate termwise.
– Lord Shark the Unknown
Nov 24 at 6:17
You can expand your formula by the binomial theorem and integrate termwise.
– Lord Shark the Unknown
Nov 24 at 6:17
If i use binomial theorem then it will consume a lot of time.Is there any other way to figure out this @Lord Shark the Unknown Sir
– emonhossain
Nov 24 at 6:18
If i use binomial theorem then it will consume a lot of time.Is there any other way to figure out this @Lord Shark the Unknown Sir
– emonhossain
Nov 24 at 6:18
As alreasy said, integrate by parts to get the reduction formula.
– Claude Leibovici
Nov 24 at 7:00
As alreasy said, integrate by parts to get the reduction formula.
– Claude Leibovici
Nov 24 at 7:00
You may look at the method I used here.
– Tianlalu
Nov 24 at 7:09
You may look at the method I used here.
– Tianlalu
Nov 24 at 7:09
|
show 2 more comments
1 Answer
1
active
oldest
votes
(Differentiation is easier than integration)
Let $f_{m,a, b}(x)=f_m=e^{ax}cos^m bx$. Take the derivative of $f_{m}$,
begin{align*}
f'_{m}&=af_{m}-bmf_{m-1}sin bx.tag{1}
end{align*}
So, if we take $g_{m}=af_{m}color{magenta}+bmf_{m-1}sin bx$, the derivative of $g_{m}$ will cancel $color{blue}{sin bx}$ out, i.e.
begin{align*}
g'_{m}&=af'_{m}hspace{8.2em}+bmbig(f'_{m-1}sin bxhspace{10.8em}+bunderbrace{f_{m-1}cos bx}_{=f_{m}}big)\
g'_{m}&=a^2f_{m}-color{blue}{abmf_{m-1}sin bx}+color{blue}{bm}big(color{blue}{af_{m-1}sin bx}-b(m-1) f_{m-2}color{orange}{sin^2 bx}+bf_mbig)\
g'_{m}&=(a^2+b^2m)f_m-b^2m(m-1)f_{m-2}color{orange}{(1-cos^2 bx)}\
g'_{m}&=(a^2+b^2m^2)f_m-b^2m(m-1)f_{m-2}.tag{2}
end{align*}
Now, we assume
$$I_{m}=int f_m~mathrm dx=int e^{ax}cos^m bx~mathrm dx,$$
integrate $(2)$, we get the desired result
$$g_m=(a^2+b^2m^2)I_{m}-b^2m(m-1)I_{m-2}.$$
Thanks @Tianlalu Really appreciated.Can i use Differentiation to get Integration in this way always?
– emonhossain
Nov 24 at 9:05
1
Yes, we can always do so once we find the appropriate $f_m$. For example, for $int cos^n x~dx$, we may choose $f_n=cos^{n-1}x sin x$ and differentiate it to cancel $sin x$ out.
– Tianlalu
Nov 24 at 9:11
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3011253%2fintegration-of-eax-cosn-bx-and-eax-sinn-bx%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
(Differentiation is easier than integration)
Let $f_{m,a, b}(x)=f_m=e^{ax}cos^m bx$. Take the derivative of $f_{m}$,
begin{align*}
f'_{m}&=af_{m}-bmf_{m-1}sin bx.tag{1}
end{align*}
So, if we take $g_{m}=af_{m}color{magenta}+bmf_{m-1}sin bx$, the derivative of $g_{m}$ will cancel $color{blue}{sin bx}$ out, i.e.
begin{align*}
g'_{m}&=af'_{m}hspace{8.2em}+bmbig(f'_{m-1}sin bxhspace{10.8em}+bunderbrace{f_{m-1}cos bx}_{=f_{m}}big)\
g'_{m}&=a^2f_{m}-color{blue}{abmf_{m-1}sin bx}+color{blue}{bm}big(color{blue}{af_{m-1}sin bx}-b(m-1) f_{m-2}color{orange}{sin^2 bx}+bf_mbig)\
g'_{m}&=(a^2+b^2m)f_m-b^2m(m-1)f_{m-2}color{orange}{(1-cos^2 bx)}\
g'_{m}&=(a^2+b^2m^2)f_m-b^2m(m-1)f_{m-2}.tag{2}
end{align*}
Now, we assume
$$I_{m}=int f_m~mathrm dx=int e^{ax}cos^m bx~mathrm dx,$$
integrate $(2)$, we get the desired result
$$g_m=(a^2+b^2m^2)I_{m}-b^2m(m-1)I_{m-2}.$$
Thanks @Tianlalu Really appreciated.Can i use Differentiation to get Integration in this way always?
– emonhossain
Nov 24 at 9:05
1
Yes, we can always do so once we find the appropriate $f_m$. For example, for $int cos^n x~dx$, we may choose $f_n=cos^{n-1}x sin x$ and differentiate it to cancel $sin x$ out.
– Tianlalu
Nov 24 at 9:11
add a comment |
(Differentiation is easier than integration)
Let $f_{m,a, b}(x)=f_m=e^{ax}cos^m bx$. Take the derivative of $f_{m}$,
begin{align*}
f'_{m}&=af_{m}-bmf_{m-1}sin bx.tag{1}
end{align*}
So, if we take $g_{m}=af_{m}color{magenta}+bmf_{m-1}sin bx$, the derivative of $g_{m}$ will cancel $color{blue}{sin bx}$ out, i.e.
begin{align*}
g'_{m}&=af'_{m}hspace{8.2em}+bmbig(f'_{m-1}sin bxhspace{10.8em}+bunderbrace{f_{m-1}cos bx}_{=f_{m}}big)\
g'_{m}&=a^2f_{m}-color{blue}{abmf_{m-1}sin bx}+color{blue}{bm}big(color{blue}{af_{m-1}sin bx}-b(m-1) f_{m-2}color{orange}{sin^2 bx}+bf_mbig)\
g'_{m}&=(a^2+b^2m)f_m-b^2m(m-1)f_{m-2}color{orange}{(1-cos^2 bx)}\
g'_{m}&=(a^2+b^2m^2)f_m-b^2m(m-1)f_{m-2}.tag{2}
end{align*}
Now, we assume
$$I_{m}=int f_m~mathrm dx=int e^{ax}cos^m bx~mathrm dx,$$
integrate $(2)$, we get the desired result
$$g_m=(a^2+b^2m^2)I_{m}-b^2m(m-1)I_{m-2}.$$
Thanks @Tianlalu Really appreciated.Can i use Differentiation to get Integration in this way always?
– emonhossain
Nov 24 at 9:05
1
Yes, we can always do so once we find the appropriate $f_m$. For example, for $int cos^n x~dx$, we may choose $f_n=cos^{n-1}x sin x$ and differentiate it to cancel $sin x$ out.
– Tianlalu
Nov 24 at 9:11
add a comment |
(Differentiation is easier than integration)
Let $f_{m,a, b}(x)=f_m=e^{ax}cos^m bx$. Take the derivative of $f_{m}$,
begin{align*}
f'_{m}&=af_{m}-bmf_{m-1}sin bx.tag{1}
end{align*}
So, if we take $g_{m}=af_{m}color{magenta}+bmf_{m-1}sin bx$, the derivative of $g_{m}$ will cancel $color{blue}{sin bx}$ out, i.e.
begin{align*}
g'_{m}&=af'_{m}hspace{8.2em}+bmbig(f'_{m-1}sin bxhspace{10.8em}+bunderbrace{f_{m-1}cos bx}_{=f_{m}}big)\
g'_{m}&=a^2f_{m}-color{blue}{abmf_{m-1}sin bx}+color{blue}{bm}big(color{blue}{af_{m-1}sin bx}-b(m-1) f_{m-2}color{orange}{sin^2 bx}+bf_mbig)\
g'_{m}&=(a^2+b^2m)f_m-b^2m(m-1)f_{m-2}color{orange}{(1-cos^2 bx)}\
g'_{m}&=(a^2+b^2m^2)f_m-b^2m(m-1)f_{m-2}.tag{2}
end{align*}
Now, we assume
$$I_{m}=int f_m~mathrm dx=int e^{ax}cos^m bx~mathrm dx,$$
integrate $(2)$, we get the desired result
$$g_m=(a^2+b^2m^2)I_{m}-b^2m(m-1)I_{m-2}.$$
(Differentiation is easier than integration)
Let $f_{m,a, b}(x)=f_m=e^{ax}cos^m bx$. Take the derivative of $f_{m}$,
begin{align*}
f'_{m}&=af_{m}-bmf_{m-1}sin bx.tag{1}
end{align*}
So, if we take $g_{m}=af_{m}color{magenta}+bmf_{m-1}sin bx$, the derivative of $g_{m}$ will cancel $color{blue}{sin bx}$ out, i.e.
begin{align*}
g'_{m}&=af'_{m}hspace{8.2em}+bmbig(f'_{m-1}sin bxhspace{10.8em}+bunderbrace{f_{m-1}cos bx}_{=f_{m}}big)\
g'_{m}&=a^2f_{m}-color{blue}{abmf_{m-1}sin bx}+color{blue}{bm}big(color{blue}{af_{m-1}sin bx}-b(m-1) f_{m-2}color{orange}{sin^2 bx}+bf_mbig)\
g'_{m}&=(a^2+b^2m)f_m-b^2m(m-1)f_{m-2}color{orange}{(1-cos^2 bx)}\
g'_{m}&=(a^2+b^2m^2)f_m-b^2m(m-1)f_{m-2}.tag{2}
end{align*}
Now, we assume
$$I_{m}=int f_m~mathrm dx=int e^{ax}cos^m bx~mathrm dx,$$
integrate $(2)$, we get the desired result
$$g_m=(a^2+b^2m^2)I_{m}-b^2m(m-1)I_{m-2}.$$
answered Nov 24 at 8:42
Tianlalu
3,12321038
3,12321038
Thanks @Tianlalu Really appreciated.Can i use Differentiation to get Integration in this way always?
– emonhossain
Nov 24 at 9:05
1
Yes, we can always do so once we find the appropriate $f_m$. For example, for $int cos^n x~dx$, we may choose $f_n=cos^{n-1}x sin x$ and differentiate it to cancel $sin x$ out.
– Tianlalu
Nov 24 at 9:11
add a comment |
Thanks @Tianlalu Really appreciated.Can i use Differentiation to get Integration in this way always?
– emonhossain
Nov 24 at 9:05
1
Yes, we can always do so once we find the appropriate $f_m$. For example, for $int cos^n x~dx$, we may choose $f_n=cos^{n-1}x sin x$ and differentiate it to cancel $sin x$ out.
– Tianlalu
Nov 24 at 9:11
Thanks @Tianlalu Really appreciated.Can i use Differentiation to get Integration in this way always?
– emonhossain
Nov 24 at 9:05
Thanks @Tianlalu Really appreciated.Can i use Differentiation to get Integration in this way always?
– emonhossain
Nov 24 at 9:05
1
1
Yes, we can always do so once we find the appropriate $f_m$. For example, for $int cos^n x~dx$, we may choose $f_n=cos^{n-1}x sin x$ and differentiate it to cancel $sin x$ out.
– Tianlalu
Nov 24 at 9:11
Yes, we can always do so once we find the appropriate $f_m$. For example, for $int cos^n x~dx$, we may choose $f_n=cos^{n-1}x sin x$ and differentiate it to cancel $sin x$ out.
– Tianlalu
Nov 24 at 9:11
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3011253%2fintegration-of-eax-cosn-bx-and-eax-sinn-bx%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
That "result" is a reduction formula, proved by integrating by parts.
– Lord Shark the Unknown
Nov 24 at 6:16
You can expand your formula by the binomial theorem and integrate termwise.
– Lord Shark the Unknown
Nov 24 at 6:17
If i use binomial theorem then it will consume a lot of time.Is there any other way to figure out this @Lord Shark the Unknown Sir
– emonhossain
Nov 24 at 6:18
As alreasy said, integrate by parts to get the reduction formula.
– Claude Leibovici
Nov 24 at 7:00
You may look at the method I used here.
– Tianlalu
Nov 24 at 7:09